
\magnification = 2000 
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros

% File Name as ATO: User Defined x y Iteration

\Title The Area Preserving Henon Twist Map.
\cl{ User Defined Example}
\LF
The User Defined entry is designed to study the behaviour of 2-dimensional maps
under forward iteration near an isolated, neutral fixed point. (We want a fixed point
inside the window since otherwise most of the iterated points will move out of sight.)
Our example is Henon's quadratic, area preserving twist map $F$:
$$
   F(x,y) := \left ( \matrix
                                         \cos(aa)\cdot x - \sin(aa)\cdot (y - e^{bb}\cdot x|x|) \\
                                         \sin(aa)\cdot x + \cos(aa)\cdot (y - e^{bb}\cdot x|x|) \\
                           \endmatrix
                 \right).
$$
Henon used $x^2$ instead of $x|x|$ for the perturbation term. See below.
\Lf
The main parameter $aa$ controls the derivative of $F$ at the fixed point $(0,0)$; 
$\hbox{d} F|_{(0,0)}$ is the rotation matrix with angle $aa$. The behaviour of the iterations
changes strongly with $aa$.
Try also $-aa$. $F$ is area preserving since the Jacobian determinant
$\det(\hbox{d} F) = 1$ everywhere. 
\Lf
By default $e^{bb}=1$. This parameter serves to choose the size of the neighborhood of the
fixed point, because of the scaling property \lf
\Ph{3}  $F(\vec{x};e^{bb}) = e^{-bb} \cdot F(e^{bb}\cdot\vec{x};1) $. \lf
We use $\exp(bb)$ instead of $bb$, because the scaling parameter is a 
multiplicative rather than an additive parameter.
\bigskip\goodbreak 
\Lf
The iteration is applied to the segment $[0,1]\cdot(cc,dd)$. The number of points on this
segment is {\it tResolution}. The default number of iterations is $ee = 2000$. The next 2000
iterations are obtained from the Action Menu Entry: {\tt Continue Curve Iteration}.   \lf
Since the graphic rendering is much slower than the the computation of iterations one can 
increase the parameter $hh$ from its default value $hh=1$ and then only one out of $hh$
iterations is shown on the screen. This is useful if one needs to see the result of a large
number of iterations. (For example $hh=4\cdot n$ in the case $aa=\pi/2$.)
\Lf
The Action Menu Entry {\tt Iterate Mouse Point Forward} allows to iterate a single point. During 
the selection the point coordinates appear on the screen. If DELETE is pressed during the
iteration then the waiting time at each step is cancelled so that the point races through its
orbit.
\Lf
The Action Menu Entry {\tt Choose Iteration Segment By Mouse} allows to Mouse-select
initial and final point of a segment on which $ff$ points will be distributed and iterated (by
default $ff=16$). The parameter $hh$ speeds up the iteration as above. After the first $ee$
iterations an Action Menu Entry is activated and allows to iterate further.
\Lf
As ususal one can {\tt translate} the image by dragging or one can {\tt scale} it  by 
depressing SHIFT and dragging vertically.
\bigskip \bigskip\goodbreak 
\Lf
One can also {\tt morph} the images. They change rather drastically with $aa$. As default
morph $bb$ is decreased so that the neighborhood of the fixed point gets expanded. One
observes that most of the iterated points travel on {\it invariant curves} around the fixed
point. Occasional periodic points clearly show up in the image. If $aa$ is an irrational 
multiple of $\pi$ then the visible periods do increase as the neighborhood of the
fixed point expands with decreasing $bb$. 
(For the default morph the number $ee$ of iterations is restricted to 500 to reduce waiting times.)
\Lf
The Henon twist map can be written as a rotation plus a quadratic perturbation:
$$\eqalign{
	&F(x,y) := \left( \matrix
	                                    \cos(aa) &-\sin(aa) \\
	                                   \sin(aa) & +\cos(aa) 
	                                   \endmatrix
	                 \right )\cdot { x\choose y}+ \hbox{perturb},                                 
}$$
  $$ \hbox{perturb} := e^{bb}\cdot x|x|\cdot {+\sin(aa) \choose -\cos(aa)}.  $$
The scalar product between the perturbation and the tangent to the rotation circles is the
$$\eqalign{
	\hbox{Forward Perturbation } &=  \cr
	-e^{bb}|x|^3\cdot(\sin^2(aa) &+ \cos^2(aa)).\hskip1cm
}$$
This explains why we changed the Henon map. Our negative forward perturbation means
that the images under $F$ stay behind the rotation image, and more so the larger $|x|$. This
is the usual behavior of a monotone twist map. Henon's perturbation has the factor $x^3$ 
instead of $|x|^3$, so  that the twist in the left half plane partially cancels the twist in the right 
half plane. In our definition do the elliptical islands around periodic points appear more
easily, while with Henon's definition the behaviour near the fixed point, in the case when $aa$ 
is a rational multiple of $\pi$ (e.g. $aa = \pi/2$), is much more complicated.
\Lf
We recommend that users try out also Henon's definition and  definitions of their own.

\bye